Hydostatics: Force and torque on a dam wall of reservoir

As the reservoir behind a dam is filled with water, the pressure that the water exerts on the dam increases. Eventually, the force on the dam becomes substantial, and it could cause the dam to collapse. There are two significant issues to be considered: First, the base of the dam should be able to withstand the pressure rho gh, where rho is the density of the water behind the dam, h is its depth, and g is the magnitude of the acceleration due to gravity. This means that the material of which the dam is made needs to be strong enough so that it doesn't crack (compressive strength).

The second issue has to do with the strength of the foundation of the dam. The water pressure exerts a clockwise torque on the dam, as shown in the figure. The foundation of the dam should be strong enough so that the dam does not topple. The material has to be strong enough that the dam does not snap (shear strength).

To study this phenomenon, consider the simple model of a dam depicted in the diagram. View Figure A reservoir of water (density rho) behind the dam is filled to a height h. Assume that the width of the dam (the dimension pointing into the screen) is L

a). Consider a horizontal layer of the dam wall of thickness dx located a distance x above the reservoir floor. What is the magnitude dF of the force on this layer due to the water in the reservoir? (in terms of x, dx, or other terms in the problem)

b) The force of the water produces a torque on the dam. In a simple model, if the torque due to the water were enough to cause the dam to break free from its foundation, the dam would pivot about its base (point P). What is the magnitude tau of the torque about the point P due to the water in the reservoir?
Express your answer in terms of quantities given in the problem introduction and the magnitude of the acceleration due to gravity g.

"Sorry I attached the wrong figure for this question but seems your answer is still right. May I resend the problem with the correct figure and let you to double check with it? If I may, how to attach the correct figure?"